CN112097677A - Rapid high-precision phase restoration method for anti-vibration interferometry - Google Patents

Rapid high-precision phase restoration method for anti-vibration interferometry Download PDF

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CN112097677A
CN112097677A CN202010870958.1A CN202010870958A CN112097677A CN 112097677 A CN112097677 A CN 112097677A CN 202010870958 A CN202010870958 A CN 202010870958A CN 112097677 A CN112097677 A CN 112097677A
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inclination
vibration
phase
interference signal
dimensional interference
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CN112097677B (en
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李建欣
段明亮
宗毅
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Nanjing University of Science and Technology
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    • G01—MEASURING; TESTING
    • G01B—MEASURING LENGTH, THICKNESS OR SIMILAR LINEAR DIMENSIONS; MEASURING ANGLES; MEASURING AREAS; MEASURING IRREGULARITIES OF SURFACES OR CONTOURS
    • G01B11/00—Measuring arrangements characterised by the use of optical techniques
    • G01B11/24—Measuring arrangements characterised by the use of optical techniques for measuring contours or curvatures
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Abstract

The invention discloses a rapid high-precision phase restoration method for anti-vibration interferometry, which comprises the following steps: taking data of a certain row and a certain column in the interference pattern to obtain two groups of one-dimensional interference signals; constructing an alternate iterative optimization model of phase distribution and inclination parameters; calculating an initial value of a tilt parameter of the one-dimensional interference signal; performing alternate iteration on the one-dimensional interference signal by using an alternate iteration optimization model to obtain a corresponding one-dimensional interference signal inclination parameter; calculating the inclination parameters of the vibration inclination plane of the original two-dimensional interferogram according to the one-dimensional inclination parameters; and solving the phase information of the original interferogram by using a least square solution method. The invention has high phase recovery precision and fast iterative convergence speed for the conditions of large vibration inclination amplitude, small number of fringes, closed fringes, non-closed fringes and non-uniform background and modulation degree, does not need to change hardware of an interference measurement system, and provides a solution with low cost and high precision for phase-shifting interference measurement in a vibration environment.

Description

Rapid high-precision phase restoration method for anti-vibration interferometry
Technical Field
The invention belongs to the field of optical interference measurement and test, and particularly relates to a rapid high-precision phase recovery method for anti-vibration interference measurement.
Background
Nowadays, optical interferometry is widely used to measure the surface shape of an optical element, and phase-shifting interferometry is the most important measurement method in interferometry due to its advantage of high precision. In phase shifting interferometry, a phase shifter is typically used to produce a phase shift of 2 π/N between interferograms, where N is 3 or greater. However, in actual measurement, the vibration of the external environment may cause random tilt error of the phase shift amount, which causes the phase shift amount to be different at different positions in the same interferogram, thereby reducing the accuracy of the measurement result.
In order to recover the measured phase from the interferogram containing the vibration tilt phase shift error, many phase extraction algorithms for processing the vibration tilt error have been developed, and these methods can be divided into two categories: one type is a direct phase extraction method, which has the characteristics of simplicity and high efficiency and can quickly extract phase information from an interferogram with a vibration tilt error, such as a space carrier frequency method, a vibration compensation algorithm and the like. However, the existing direct calculation method is generally low in precision and is not enough to meet the requirement of high-precision measurement. The second type is an iterative method, which does not need to obtain phase shift information in advance, uses the vibration tilt phase shift as an unknown quantity, and performs alternate iterative optimization with the measured phase so as to solve the tilt phase shift quantity and the measured phase at the same time. However, most of the existing iteration methods use mathematical approximation of first-order taylor expansion to establish a linear least square iteration model, or use a nonlinear least square method to iterate, and such iteration methods have mathematical approximation errors and are easy to fall into a local optimal solution, so that only interferograms with small vibration tilt amplitudes can be processed. Whether a direct calculation method or an iterative method is adopted, the performances of the methods are greatly influenced under the conditions of large vibration inclination amplitude, small stripe number (for example, as few as one stripe), closed circular stripes, non-uniform background and modulation degree and the like.
Disclosure of Invention
The invention aims to provide a rapid high-precision phase restoration method for anti-vibration interference measurement, which can inhibit interference phase errors caused by vibration and improve the measurement precision.
The technical solution for realizing the purpose of the invention is as follows: a method for fast high precision phase retrieval for anti-vibration interferometry, the method comprising the steps of:
step 1, aiming at each interference pattern In(x,y)，Respectively extracting ith row and jth column data to obtain two groups of one-dimensional interference signals In(x, I) and In(j,y)；
Step 2, constructing an alternate iterative optimization model of phase distribution and inclination parameters aiming at the one-dimensional interference signals;
step 3, calculating initial values of inclination parameters of the two groups of one-dimensional interference signals;
and 4, respectively carrying out alternate iteration on the two groups of one-dimensional interference signals by using an alternate iteration optimization model based on the initial value of the inclination parameter to obtain the corresponding one-dimensional interference signal inclination parameter (alpha)n,n,i) And (beta)n,n,j)；
Step 5, according to the one-dimensional interference signal inclination parameter (alpha)n,n,i) And (beta)n,n,j) Calculating the original interferogram InInclination parameter alpha of (x, y) vibration inclination planen，βnAndn；
step 6, based on the inclination parameter alphan，βnAndnsolving the original interferogram I by using a least square solution methodnPhase information phi (x, y) of (x, y).
Compared with the prior art, the invention has the following remarkable advantages: 1) a linear least square alternative iteration optimization model about phase distribution and inclination parameters is established, and a mathematical approximation method and a non-least square method are not needed, so that the condition of large vibration inclination amplitude can be solved; 2) the inclination amount is obtained by adopting two processes of initial value estimation and fine iteration based on an alternating iteration optimization algorithm frame, so that the problem of falling into a local optimal solution is avoided, and the robustness of phase restoration is improved; 3) the vibration inclined plane is solved through one-dimensional interference signals in two orthogonal directions, and has the advantages of low complexity and small operand; 4) the method has high recovery precision for the conditions of large vibration amplitude, small number of stripes, closed and non-closed stripes, and uneven background and modulation degree.
The present invention is described in further detail below with reference to the attached drawing figures.
Drawings
FIG. 1 is a flow chart of the fast high-precision phase recovery method for anti-vibration interferometry of the present invention.
FIG. 2 is 4 interferograms used for the calculations in one embodiment.
FIG. 3 is a diagram illustrating a phase distribution obtained by the method of the present invention according to an embodiment.
FIG. 4 is a diagram illustrating a phase distribution obtained by a conventional four-step phase shifting method according to an embodiment.
FIG. 5 is a diagram illustrating the distribution of phase residuals obtained by the method and the synchronous phase shift measurement according to an embodiment of the present invention.
Detailed Description
In order to make the objects, technical solutions and advantages of the present application more apparent, the present application is described in further detail below with reference to the accompanying drawings and embodiments. It should be understood that the specific embodiments described herein are merely illustrative of the present application and are not intended to limit the present application.
In one embodiment, in conjunction with fig. 1, there is provided a method for fast high precision phase recovery for anti-vibration interferometry, the method comprising the steps of:
step 1, aiming at each interference pattern In(x, y), extracting the ith row and jth column data to obtain two groups of one-dimensional interference signals In(x, I) and In(j,y)；
Step 2, constructing an alternate iterative optimization model of phase distribution and inclination parameters aiming at the one-dimensional interference signals;
step 3, calculating initial values of inclination parameters of the two groups of one-dimensional interference signals;
and 4, respectively carrying out alternate iteration on the two groups of one-dimensional interference signals by using an alternate iteration optimization model based on the initial value of the inclination parameter to obtain the corresponding one-dimensional interference signal inclination parameter (alpha)n,n,i) And (beta)n,n,j)；
Step 5, according to the one-dimensional interference signal inclination parameter (alpha)n,n,i) And (beta)n,n,j) Calculating the original interferogram InInclination parameter alpha of (x, y) vibration inclination planen，βnAndn；
step 6, based on the inclination parameter alphan，βnAndnsolving the original interferogram I by using a least square solution methodnPhase information phi (x, y) of (x, y).
Further, in one embodiment, the optimization process of the alternating iterative optimization model in step 2 specifically includes:
(1) for a certain one-dimensional interference signal In(x)
One-dimensional interference signal In(x) The light intensity expression of (a) is:
In(x)＝a(x)+b(x)cos(φ(x)+pn(x))
where N is the interferogram number, N is 1,2, N is the total number of interferograms, and X is [1,2, X]Is a coordinate vector, X is the total number of sampling points, a (X), b (X), phi (X) are respectively a background item, a modulation item and the phase distribution to be measured; p is a radical ofn(x)＝αnx+nIs a one-dimensional interference signal In(x) Amount of vibration phase inclination of alphanAndnare all tilt parameters, whereinnIn order to be the tilt factor,nis a phase shift quantity;
step 2-1, interference signal In(x) The light intensity expression of (a) is rewritten as:
In(x)＝a(x)+c(x)cos(pn(x))+s(x)sin(pn(x))
wherein c (x) is (b), (x) cos (Φ (x)), and s (x) is (b) (x) sin (Φ (x));
using least squares solution, using known
Figure BDA0002651061220000031
And
Figure BDA0002651061220000032
calculating a (x), b (x), and φ (x):
Figure BDA0002651061220000033
the above-mentioned
Figure BDA0002651061220000041
And
Figure BDA0002651061220000042
representing the inclination parameters obtained by the k iteration; here, a (x) is directly calculated by the least squares solution method;
step 2-2, constructing and solving a linear equation set of the vibration phase inclination quantity by a space and time related variable decoupling method, and utilizing the known method
Figure BDA0002651061220000043
And a (x), b (x) and phi (x) obtained in the step 2-1, and calculating by combining a linear least square method
Figure BDA0002651061220000044
And
Figure BDA0002651061220000045
step 2-3, repeating steps 2-1 and 2-2, and performing alternate iterative optimization between phase distribution and inclination parameters until alpha is reachednAndnconvergence is achieved by the following calculation formula:
Figure BDA0002651061220000046
in the formula (I), the compound is shown in the specification,
Figure BDA0002651061220000047
1and2and setting the two values in a self-defined mode according to actual requirements for convergence threshold values of two adjacent iterations.
(2) For another one-dimensional interference signal In(y)
One-dimensional interference signal InThe light intensity expression of (y) is:
In(y)＝a(y)+b(y)cos(φ(y)+pn(y))
where N is the interferogram number, N is 1,2, N is the total number of interferograms, y is [ 1%,2,···,Y]Is a coordinate vector, Y is the total number of sampling points, a (Y), b (Y) and phi (Y) are respectively a background item, a modulation item and phase distribution to be measured; p is a radical ofn(y)＝βny+nIs a one-dimensional interference signal In(y) amount of inclination of vibration phase, βnAndnare all tilt parameters, whereinnIn order to be the tilt factor,nis a phase shift quantity;
step 2-1', interference signal InThe light intensity expression of (y) is rewritten as:
In(y)＝a(y)+c(y)cos(pn(y))+s(y)sin(pn(y))
wherein c (y) is b (y) cos (Φ (y)), s (y) is b (y) sin (Φ (y));
using least squares solution, using known
Figure BDA0002651061220000048
And
Figure BDA0002651061220000049
calculating a (y), b (y), and φ (y):
Figure BDA00026510612200000410
the above-mentioned
Figure BDA00026510612200000411
And
Figure BDA00026510612200000412
representing the inclination parameters obtained by the k iteration;
step 2-2', a linear equation set for solving the vibration phase inclination quantity is constructed by a method of space and time related variable decoupling, and the known method is utilized
Figure BDA0002651061220000051
And a (y), b (y) and phi (y) obtained in the step 2-1', and calculating by combining a linear least square method
Figure BDA0002651061220000052
And
Figure BDA0002651061220000053
step 2-3 ', repeating steps 2-1 ' and 2-2 ', and performing alternate iterative optimization between phase distribution and inclination parameters until betanAndnconvergence is achieved by the following calculation formula:
Figure BDA0002651061220000054
in the formula (I), the compound is shown in the specification,
Figure BDA0002651061220000055
1and2and setting the two values in a self-defined mode according to actual requirements for convergence threshold values of two adjacent iterations.
Further, in one embodiment, the method of decoupling the spatially and temporally dependent variables described in step 2-2 constructs a linear system of equations for solving the amount of phase tilt, and utilizes known techniques
Figure BDA0002651061220000056
And a (x), b (x) and phi (x) obtained in the step 2-1, and calculating by combining a linear least square method
Figure BDA0002651061220000057
And
Figure BDA0002651061220000058
the specific process comprises the following steps:
step 2-2-1, converting p into u ═ x + u by vector transformationn(x) And phi (x) are decoupled, and the vibration phase inclination quantity p independent of the space coordinate is obtainedn(u) is:
pn(u)＝αnu+n
wherein U is 0,1, …, U is an integer, and U is less than X;
step 2-2-2, combining said vector transformation, tilt independent of spatial coordinatesSkew amount and one-dimensional interference signal In(x) The interference light intensity in the vector region u is obtained as follows:
In(u)＝a(u)+b(u)cos(φ(u)+αnx+pn(u))
step 2-2-3, constructing a linear equation system for solving the vibration phase inclination amount:
Figure BDA0002651061220000059
in the formula, cn(u)＝cos(pn(u))，sn(u)＝-sin(pn(u))，
Figure BDA00026510612200000510
Step 2-2-4, utilizing known
Figure BDA00026510612200000511
And a (x), b (x) and phi (x) obtained in the step 2-1, and obtaining p by a linear least square methodn(u) calculated from
Figure BDA00026510612200000512
And
Figure BDA00026510612200000513
Figure BDA00026510612200000514
Figure BDA0002651061220000061
in the formula,. DELTA.pn(u)＝pn(u+1)-pn(u)。
Further, the method of decoupling through space and time dependent variables in step 2-2' constructs a linear equation system for solving the vibration phase inclination amount, and utilizes the known method
Figure BDA0002651061220000062
And a (y), b (y) and phi (y) obtained in the step 2-1', and calculating by combining a linear least square method
Figure BDA0002651061220000063
And
Figure BDA0002651061220000064
the method specifically comprises the following steps:
step 2-2' -1, converting p into y + u by vector transformationn(y) and phi (y) are decoupled to obtain a vibration phase tilt amount p independent of the spatial coordinatesn(u) is:
pn(u)＝βnu+n
wherein U is 0,1, …, U is an integer, and U is less than Y;
step 2-2' -2, combining said vector transformation, the amount of tilt independent of the spatial coordinates and the one-dimensional interference signal In(y) to obtain an interference intensity in the vector area u as:
In(u)＝a(u)+b(u)cos(φ(u)+βny+pn(u))
step 2-2' -3, constructing a linear equation system for solving the vibration phase inclination:
Figure BDA0002651061220000065
in the formula, cn(u)＝cos(pn(u))，sn(u)＝-sin(pn(u))，
Figure BDA0002651061220000066
Step 2-2' -4, using known
Figure BDA0002651061220000067
And a (y), b (y) and phi (y) obtained in the step 2-1', and obtaining p by a linear least square methodn(u) calculated from
Figure BDA0002651061220000068
And
Figure BDA0002651061220000069
Figure BDA00026510612200000610
Figure BDA00026510612200000611
in the formula,. DELTA.pn(u)＝pn(u+1)-pn(u)。
Further, in one embodiment, the step 3 of calculating the initial values of the tilt parameters of the two sets of one-dimensional interference signals includes:
step 3-1, dividing each dimension interference signal into M subsections with length of l (wherein, when M is not an integer, M is rounded downwards), wherein the interval of adjacent subsections is d, d is more than 0 and less than l, and the subsections are partially overlapped to obtain M time sequence interference subsections sequences; the sub-segments are small enough, and the sub-segments on the time sequence only have phase shift without inclination;
step 3-2, setting the following parameters:
Figure BDA0002651061220000071
the initial value of the phase shift quantity of the mth time sequence interference sub-section sequence is the phase shift quantity sigma of the last time sequence interference sub-section sequencen(m-1), U ═ 0; then, the phase shift quantity sigma of the mth time sequence interference subsegment sequence is obtained by using the alternating iteration optimization model of the step 2n(m)，m＝0,1,···,M；
Step 3-3, using the phasor σn(m) calculating initial values of two one-dimensional interference signal tilt parameters
Figure BDA0002651061220000072
And
Figure BDA0002651061220000073
the calculation formulas are respectively as follows:
Figure BDA0002651061220000074
Figure BDA0002651061220000075
in the formula, Δ σn(m)＝σn(m+1)-σn(m)。
Further, in one embodiment, the step 5 is based on the one-dimensional interference signal inclination parameter (α)n,n,i) And (beta)n,n,j) Calculating the original interferogram InInclination parameter alpha of (x, y) vibration inclination planen，βnAndnthe method specifically comprises the following steps:
original interferogram InThe expression for the plane of vibration tilt of (x, y) is: p is a radical ofn(x,y)＝αnx+βny+n；
Wherein alpha in the one-dimensional interference signal tilt parameternIs pnInclination parameter alpha of (x, y)n；
Beta in one-dimensional interference signal tilt parameternIs pnTilt parameter β of (x, y)n；
nThe calculation formula of (2) is as follows:
n＝(n,i-βni)/2+(n,j-αnj)/2。
as a specific example, in one embodiment, the verification and description of the high-precision phase restoration method for anti-vibration interferometry of the present invention specifically includes:
in this embodiment, a PhaseCam 6000 dynamic interferometer is used to acquire an interferogram of a spherical mirror with a caliber of 50mm and an F number of 5 in a vibration environment. The method is characterized in that 15 interferograms are continuously acquired under a vibration environment, the size of the interferograms is 1008 pixels by 1008 pixels, and each interferogram can be decomposed into 4 sub-interferograms with 504 pixels by 504 pixels and pi/2 phase shift. The sub-interferograms of the same polarization state at 4 different moments are shown in fig. 2, and the method of the invention is adopted for recovering the phase to be detected for 15 sub-interferograms, and the steps are as follows:
step 1, extracting 15 interferogram sequences In(x, y) in the 252 th row and the 252 th column, two groups of one-dimensional interference signals I are obtainedn(x,252) and In(252,y)；
Step 2, based on space and time related variable decoupling, establishing a linear least square alternative iteration optimization model related to phase distribution and inclination parameters, and setting convergence thresholds in steps 2-3 as1＝10-5And2＝10-3；
step 3, respectively carrying out initial value estimation of the vibration inclination coefficient on the two groups of one-dimensional interference signals by adopting an overlapping segmentation iteration method to obtain the initial value of the inclination coefficient of the two groups of interference signals
Figure BDA0002651061220000081
And
Figure BDA0002651061220000082
setting the sub-segment length l as 128 and the sub-segment interval d as 32 in the step 3-1;
step 4, taking the result obtained in the step 3 as an iteration initial value, and respectively carrying out fine iteration on the two groups of one-dimensional interference signals by adopting an alternative iteration optimization model to obtain a high-precision vibration inclination parameter (alpha)n,n,i) And (beta)n,n,j)；
Step 5, calculating the original interferogram I through the inclination parameters of the one-dimensional signals obtained in the step 4n(x, y) vibration tilt phase plane coefficient αn，βnAndn；
step 6, based on the inclination parameter alphan，βnAndnsolving the original interferogram I by using a least square solution methodnThe phase information phi (x, y) of (x, y) is shown in fig. 3.
The acquired interferogram is calculated by a conventional four-step phase shifting method, the obtained phase distribution is shown in fig. 4, and it can be seen that the result has obvious double-frequency ripple, which indicates that vibration can bring about a large interference term as an error. In contrast, the method of the present invention results in no significant ripple error distribution. In addition, compared with the calculation result of the synchronous phase shift measurement of the 4D dynamic interferometer, the method has high consistency, the phases of the two residual errors are shown in FIG. 5, and the phase residual errors PV and RMS are very small and are respectively 0.0420 lambda and 0.0046 lambda. Therefore, the method has good anti-vibration performance, and can effectively inhibit phase errors caused by vibration on the actual interference pattern with large vibration inclination amplitude, small number of fringes and closed fringes.
The foregoing illustrates and describes the principles, general features, and advantages of the present invention. It will be understood by those skilled in the art that the present invention is not limited to the embodiments described above, which are described in the specification and illustrated only to illustrate the principle of the present invention, but that various changes and modifications may be made therein without departing from the spirit and scope of the present invention, which fall within the scope of the invention as claimed. The scope of the invention is defined by the appended claims and equivalents thereof.

Claims (5)

1. A method for fast high precision phase retrieval for anti-vibration interferometry, comprising the steps of:
step 1, aiming at each interference pattern In(x, y), extracting the ith row and jth column data to obtain two groups of one-dimensional interference signals In(x, I) and In(j,y)；
Step 2, constructing an alternate iterative optimization model of phase distribution and inclination parameters aiming at the one-dimensional interference signals;
step 3, calculating initial values of inclination parameters of the two groups of one-dimensional interference signals;
and 4, respectively carrying out alternate iteration on the two groups of one-dimensional interference signals by using an alternate iteration optimization model based on the initial value of the inclination parameter to obtain the corresponding one-dimensional interference signal inclination parameter (alpha)n,n,i) And (beta)n,n,j)；
Step 5, according to the one-dimensional interference signal inclination parameter (alpha)n,n,i) And (beta)n,n,j) Calculating the original interferogram InInclination parameter alpha of (x, y) vibration inclination planen，βnAndn；
step 6, based on the inclination parameter alphan，βnAndnsolving the original interferogram I by using a least square solution methodnPhase information phi (x, y) of (x, y).
2. The method for fast high-precision phase recovery for anti-vibration interferometry according to claim 1, wherein the optimization procedure of the alternating iterative optimization model in step 2 specifically comprises:
(1) for a certain one-dimensional interference signal In(x)
One-dimensional interference signal In(x) The light intensity expression of (a) is:
In(x)＝a(x)+b(x)cos(φ(x)+pn(x))
where N is the interferogram number, N is 1,2, N is the total number of interferograms, and X is [1,2, X]Is a coordinate vector, X is the total number of sampling points, a (X), b (X), phi (X) are respectively a background item, a modulation item and the phase distribution to be measured; p is a radical ofn(x)＝αnx+nIs a one-dimensional interference signal In(x) Amount of vibration phase inclination of alphanAndnare all tilt parameters, whereinnIn order to be the tilt factor,nis a phase shift quantity;
step 2-1, interference signal In(x) The light intensity expression of (a) is rewritten as:
In(x)＝a(x)+c(x)cos(pn(x))+s(x)sin(pn(x))
wherein c (x) is (b), (x) cos (Φ (x)), and s (x) is (b) (x) sin (Φ (x));
using least squares solution, using known
Figure FDA0002651061210000021
And
Figure FDA0002651061210000022
calculating a (x), b (x), and φ (x):
Figure FDA0002651061210000023
φ(x)＝tan-1(-s(x)/c(x))
the above-mentioned
Figure FDA0002651061210000024
And
Figure FDA0002651061210000025
representing the inclination parameters obtained by the k iteration;
step 2-2, constructing and solving a linear equation set of the vibration phase inclination quantity by a space and time related variable decoupling method, and utilizing the known method
Figure FDA0002651061210000026
And a (x), b (x) and phi (x) obtained in the step 2-1, and calculating by combining a linear least square method
Figure FDA0002651061210000027
And
Figure FDA0002651061210000028
step 2-3, repeating steps 2-1 and 2-2, and performing alternate iterative optimization between phase distribution and inclination parameters until alpha is reachednAndnconvergence is achieved by the following calculation formula:
Figure FDA0002651061210000029
in the formula (I), the compound is shown in the specification,
Figure FDA00026510612100000210
1and2a convergence threshold for two adjacent iterations;
(2) for another one-dimensional interference signal In(y) process and interference signal In(x) The process of (a) is the same, except that: all x are replaced by y, αnSubstitution to betan，pn(y)＝βny+n。
3. The fast high-precision phase recovery method for anti-vibration interferometry according to claim 2, wherein step 2-2 constructs a linear equation system for solving the phase tilt amount by a method of spatial and temporal dependent variable decoupling and using known methods
Figure FDA00026510612100000211
And a (x), b (x) and phi (x) obtained in the step 2-1, and calculating by combining a linear least square method
Figure FDA00026510612100000212
And
Figure FDA00026510612100000213
the specific process comprises the following steps:
step 2-2-1, converting p into u ═ x + u by vector transformationn(x) And phi (x) are decoupled, and the vibration phase inclination quantity p independent of the space coordinate is obtainedn(u) is:
pn(u)＝αnu+n
wherein U is 0,1, …, U is an integer, and U is less than X;
step 2-2-2, combining the vector transformation, the inclination quantity irrelevant to the space coordinate and the one-dimensional interference signal In(x) The interference light intensity in the vector region u is obtained as follows:
In(u)＝a(u)+b(u)cos(φ(u)+αnx+pn(u))
step 2-2-3, constructing a linear equation system for solving the vibration phase inclination amount:
Figure FDA0002651061210000031
in the formula, cn(u)＝cos(pn(u))，sn(u)＝-sin(pn(u))，
Figure FDA0002651061210000032
Step 2-2-4, utilizing known
Figure FDA0002651061210000033
And a (x), b (x) and phi (x) obtained in the step 2-1, and obtaining p by a linear least square methodn(u) calculated from
Figure FDA0002651061210000034
And
Figure FDA0002651061210000035
Figure FDA0002651061210000036
Figure FDA0002651061210000037
in the formula,. DELTA.pn(u)＝pn(u+1)-pn(u)。
4. The method for fast high-precision phase recovery for anti-vibration interferometry according to claim 3, wherein the step 3 of calculating the initial values of the tilt parameters of the two sets of one-dimensional interference signals comprises:
step 3-1, dividing each dimension interference signal into M subsections with the length of l, wherein the interval between every two adjacent subsections is d, d is more than 0 and less than l, and the subsections are partially overlapped to obtain M time sequence interference subsections sequences;
step 3-2, setting the following parameters:
Figure FDA0002651061210000038
the initial value of the phase shift quantity of the mth time sequence interference sub-section sequence is the phase shift quantity sigma of the last time sequence interference sub-section sequencen(m-1), U ═ 0; then, the phase shift quantity sigma of the mth time sequence interference subsegment sequence is obtained by using the alternating iteration optimization model of the step 2n(m)，m＝0,1,···,M；
Step 3-3, using the phasor σn(m) calculating initial values of two one-dimensional interference signal tilt parameters
Figure FDA0002651061210000039
And
Figure FDA00026510612100000310
the calculation formulas are respectively as follows:
Figure FDA00026510612100000311
Figure FDA00026510612100000312
in the formula, Δ σn(m)＝σn(m+1)-σn(m)。
5. Method for fast and high-precision phase recovery for anti-vibration interferometry according to claim 4, characterized in that step 5 said tilting parameter (α) according to said one-dimensional interference signaln,n,i) And (beta)n,n,j) Calculating the original interferogram InInclination parameter alpha of (x, y) vibration inclination planen，βnAndnthe method specifically comprises the following steps:
original interferogram InThe expression for the plane of vibration tilt of (x, y) is: p is a radical ofn(x,y)＝αnx+βny+n；
Wherein, the one-dimensional interference signal is inclinedAlpha in the amountnIs pnInclination parameter alpha of (x, y)n；
Beta in one-dimensional interference signal tilt parameternIs pnTilt parameter β of (x, y)n；
nThe calculation formula of (2) is as follows:
n＝(n,i-βni)/2+(n,j-αnj)/2。
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